1) Yes. It will be approved within a day. However, it will be void on the day of travel (explained further down)
2) You can take a rapid antigen test for USA still. It must be taken within three days of travel to USA, not your flight to Heathrow and you must have a certificate. Full testing requirements at the bottom.
3) I am not sure regarding the requirement to show vaccination etc. When I went in June Pennsylvania had a mandate that if you were vaccinated you didn't need to wear a mask. I was never asked but I was just going to show my NHS app if ever asked. However, when in shops etc I chose to wear a mask anyway. At airports and on aircraft you must always have a mask worn, excluding eating and drinking.
4) Yes. You must bring your marriage certificate which is your husband's proof that he is exempt from the proclamation. If you do not bring this he will be denied entering the USA under any circumstances. You also need the attestation which is essentially a form to claim that your test result is true. You need one each. These can be filled out at check-in but TRUST ME, you do not want that added stress. So, you will need: - marriage certificate, - test results, - attestation. On top of your regular travel documents and forms.
5) Can't answer. Depends on your state's self-isolation rules. They're all different. All I'll say is 'must and shall' is different to 'recommend and advise'.
What to expect at Heathrow
Your husband will be checked by a US immigration officer that he is exempt from the proclamation by presenting his evidence (i.e. marriage certificate). He will then be cleared to proceed to check-in. This can take a long time. There is usually only one immigration officer who has three phones all calling him from boarding gates, different areas, USA border control etc. At check-in, you present your test certificates, attestation and passports. Your husband's ESTA will be cancelled at this stage. Check-in staff then call US border control and say your husband is exempt from the proclamation and they will then send a code to the check-in staff to manually override and reinstate the ESTA. You'll then be checked in and good to go. This process took me, my wife and daughter about 1 hour. Ensure you get there in plenty of time. Our gate was also probably in the furthest part of Heathrow also where you need to get on a train to get to. Getting there 2-3 hours before can quickly turn into 'oh god, we're gonna have to run' so don't be casual after checking in. Also, DO NOT do online check-in. Leave all info off the flight website. Your husband must be manually checked in due to the above. Checking in online will cause all types of problems.
Official CDC testing requirements
What types of SARS-CoV-2 test are acceptable under the Order?
Passengers must be tested with a viral test that could be either an antigen test or a nucleic acid amplification test (NAAT). Examples of available NAATs for SARS-CoV-2 include but are not restricted to reverse transcription polymerase chain reaction (RT-PCR), reverse transcription loop-mediated isothermal amplification (RT-LAMP), transcription-mediated amplification (TMA), nicking enzyme amplification reaction (NEAR), and helicase-dependent amplification (HDA). The test used must be authorized for use by the relevant national authority for the detection of SARS-CoV-2 in the country where the test is administered. A viral test conducted for U.S. Department of Defense (DOD) personnel, including DOD contractors, dependents, and other U.S. government employees, and tested by a DOD laboratory located in a foreign country also meets the requirements of the Order.